1. For the Results related to Multivariate Analysis :(1) The first stage fundamentals for the sampling theory under a elliptical population : When the population is ellipitical, sample mean x^^- and sample covariance matrix S are not necessarily independent, which is different from the Normal case. This causes the difficulties in the analysis. Overcoming of these difficulties is essential in the development of the theory.In the research, formulas for obtaining the limiting distribution of functions of S, f(S) and asymptotic expansion for E[f(S)] are given. The concrete example are given. The research may promise further fruits for the multiple comparisons.(2) In the normal case, the modified second approximation method for evaluating the upper percentiles of T^2_-type statistics is examined by large scale of Monte Carlo numerical experiments. This is done concretely for the multidimensional range and for the statistics defined for comparing several trial treatments with a control treat
… Morement. Also effective range of parameters sufficient for the parctical use is determined.2. For the Results of Experimental Designs and Analysis of Variance :Convariance matrix SIGMA in the general linear model x - N_p(R_Xl_p + mu, SIGMA ) is of some specific structure. Under this structure of SIGMA, the sample space is decomposed, with respect to SIGMA, into latent subspaces corresponding to distinct latent roots. Correspondingly the overall hypothesis that all components of mu are equal is decomposed into sets of subhypotheses of multiple contrasts c_i'mu = 0, i=1, 2, ..., r. And F-statistics corresponding to subhypothese are defined. However it should be checked, in this mathematical formation, whether or not the setted subhypotheses have practical meanings. In this project, cases of split plot design, repeated observations, and cyclic random vector are investigated concretely. Fortunately, in these special cases, reducted subhypotheses have practical meaning when the dimensionaliy is lower values (p=1, 2, ..., 5). Finally these test procedures are compared with the well known Hotelling's T^2-test procedure with respect to effectiveness. Less